Effective dynamics from minimizing dissipation

It is known that the same physical system can be described by different effective theories depending on the scale at which it is observed. In this work, we formulate a prescription for finding the unitary that best approximates the large-scale dynamics of a quantum system evolving discretely in time, as it is the case for digital quantum simulators. We consider the situation in which the degrees of freedom of the system can be divided between an IR part that we can observe and a UV part that we cannot observe. Following a principle of minimal dissipation, our goal is to find the unitary dynamics that best approximates the (generally nonunitary) time evolution of the IR degrees of freedom. We first prove that when the IR and UV degrees of freedom are weakly coupled, the unitary that maximizes the fidelity is given by a mean-field dynamics and the error is given by a sum of energy variances. We then apply our results to a one-dimensional quantum walk, which is known to reproduce the Dirac equation in the small-mass and -momentum limit. We find that in this limit the effective IR dynamics is obtained by a mass redefinition.